Constantin C. Brîncuş

March1st until August 31st, 2025

Affiliation: Institute of Philosophy and Psychology, Romanian Academy // Faculty of Philosophy, University of Bucharest

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Research for a study about:

From Hilbert to Carnap and beyond: Categoricity by Inferential Conservativity

The main objective of this project is to develop a new solution to the problem of categoricity, i.e. of uniquely determining by proof-theoretic means the standard model of a formal theory. Unlike other recent proposals, my solution draws on the historical and philosophical background of Hilbert’s formalism and the Vienna Circle logical empiricism.
Due to Gödel’s results, the categoricity of a theory and the semantic completeness of its underlying logic are actually mutually exclusive. In second-order logic (SOL) we obtain the categoricity of Peano Arithmetic (PA), but SOL with standard models (in which PA is categorical) is semantically incomplete. In first-order logic we have semantic completeness, but all first-order theories that allow infinite models are non-categorical. Moreover, the syntactic completeness of PA is lost in both logics. One way to obtain syntactic completeness is by using Hilbert’s new rule of inference, i.e. the ω-rule, as Carnap did. The omega logic is semantically complete and PA with omega logic (PAω) is syntactically complete. What is still lost is the categoricity of PA.
              One way to obtain the categoricity of PAω is by adding supplementary constraints. The argument for the categoricity of PAω that I aim to develop uses a proof-theoretic notion of conservativity that goes back to Carnap, Belnap, Dummett and Brandom, i.e. inferential conservativity. In this sense, a theory or a system of rules is inferentially conservative if and only if no extension of the original language allows new (material) inferences in the old system. This notion of conservativity is inferential since it assumes that the meaning of an expression is determined by the inferential use of that expression. The material inferences are those that determine the meanings of the non-logical expressions. The argument that I want to develop aims to establish that if PAω is inferentially conservative, then it is categorical.